100% right fairy.
I went into my PhD program, letting my PI know my intentions, and he supported them. I asked him, and he agreed, to write in my LoR that he knew I wanted vet school, and whole heartily supported the decision.
As far as the "Why did you do a grad program", I think that is easier to get around. "Sir, Vet school has always been my dream, I did poorly in undergrad for (whatever reasons), but I was determined to show I have improved".
"uh, I meant I want to go into industry and be a lab animal research vet".
Whatever route you choose, you need to know your own background story for the adcoms, and support it.
Finally, some vet schools will outright NOT take you if you are not finished or finishing the degree at the time of admissions. Many will, out of support for your grad school, not let you 'drop out' just to go to vet school.